Obsidian is formed from lava, and I think can only ever be 1 deep (naturally) - so if the original lava was more than 1 deep (it often is, except perhaps at edges) then there will be lava under obsidian.
So in summary you'll need to replace the lava below it with something else before mining it - place a solid block and it removes the lava in that square.
Use torches and/or signs to mark your way underground, and help keep track of where you are.
Though some of the cave systems are so large and sprawling you'll still get lost, but that's half the fun sometimes. :P
I found a room underground with a load of mossy stone, which I thought was rare, but there were also two chests containing a few little stacks of random stuff and a pig spinning in a cage (really). Is this normal?
That, sir, is a dungeon. If we weren't playing on Peaceful then that weird thing in the middle would be spawning monsters. It is called a spawner. It has a pig in it, rather than a monster, because we are on peaceful.
The chests (and the stuff in them) are your reward for finding/neutralising it!
If it weren't on peaceful, you can stop a spawner from spawning by placing torches near it after fighting your way in.
South West of the under-construction docks, South East of the partially-built railway station, North-by-NorthWest of the island which looks like a chopped-in-half christmas tree.
Yes - it's still relatively rare, but not nearly so much as before.
I think I needed two and a half deposits worth to get enough.
Not quite enough. There's a lot of naked blue sheep wandering around, but I still need at least five times more than what I've got so far. :P
(So if anyone has any Lapis Lazuli they don't need...)